Find the zeroes of the quadratic polynomial $3x^{2}-75$ and verify the relationship between the zeroes and the coefficients.

Given: The quadratic polynomial $3x^{2}-75=0$.

To do: To find the zeroes of the given quadratic polynomial and to and verify the relationship between the zeroes and the coefficients.

Solution:

Given quadratic polynomial is

$3x^{2}-75=0$

$\Rightarrow 3(x ^{2}  - 25) = 0$

$\Rightarrow 3((x) ^{2}  - (5) ^{2} ) = 0$

$\Rightarrow 3(x + 5)(x  - 5) = 0$

$\Rightarrow (x + 5)(x - 5) = 0$

Either $x + 5 =0$

$\Rightarrow x =- 5$

Or $x -5  = 0$

$\Rightarrow x =5$

Thus, $x=-5,\ 5$

On comparing the given quadratic polynomial to $ax^{2}+bx+c=0$, we have $a=3,\ b=0\ and\ c=75$ 

Sum of zeroes $=\frac{ - b}{a}$  

$\Rightarrow 5 +( - 5)=\frac{0}{3}$  

$\Rightarrow 0 = 0$  

Product of zeroes $=\frac{c}{a}$  

$\Rightarrow 5\times( - 5)=\frac{ - 75}{3}$  

$\Rightarrow - 25=- 25$